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Algebra by Serge Lang
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Algebra (1965)

by Serge Lang

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Indeholder "Foreword", "Logical Prerequisites", "Bibliography", "Part One. Groups, Rings, and Modules", "Chapter I, Groups", " 1. Monoids", " 2. Groups", " 3. Cyclic groups", " 4. Normal subgroups", " 5. Operations of a group on a set", " 6. Sylow subgroups", " 7. Categories and functors", " 8. Free groups", " 9. Direct sums and free abelian groups", " 10. Finitely generated abelian groups", " 11. The dual group", "Chapter II, Rings", " 1. Rings and homomorphisms", " 2. Commutative rings", " 3. Localization", " 4. Principal rings", "Chapter III, Modules", " 1. Basic definitions", " 2. The group of homomorphisms", " 3. Direct products and sums of modules", " 4. Free modules", " 5. Vector spaces", " 6. The dual space", "Chapter IV, Homology", " 1. Complexes", " 2. Homology sequence", " 3. Euler characteristic", " 4. The Jordan-Hölder theorem", "Chapter V, Polynomials", " 1. Free algebras", " 2. Definition of polynomials", " 3. Elementary properties of polynomials", " 4. The Euclidian algorithm", " 5. Partial fractions", " 6. Unique factorization in several variables", " 7. Criteria for irreducibility", " 8. The derivative and multiple roots", " 9. Symmetric polynomials", " 10. The resultant", "Chapter VI, Noetherian Rings and Modules", " 1. Basic criteria", " 2. Hilbert's theorem", " 3. Power series", " 4. Associated primes", " 5. Primary decomposition", "Part Two, Field Theory", "Chapter VII, Algebraic Extensions", " 1. Finite and algebraic extensions", " 2. Algebraic closure", " 3. Splitting fields and normal extensions", " 4. Separable extensions", " 5. Finite fields", " 6. Primitive elements", " 7. Purely inseparable extensions", "Chapter VIII, Galois Theory", " 1. Galois extensions", " 2. Examples and applications", " 3. Roots of unity", " 4. Linear independence of characters", " 5. The norm and trace", " 6. Cyclic extensions", " 7. Solvable and radical extensions", " 8. Kummer theory", " 9. The equation X^n - a = 0", " 10. Galois cohomology", " 11. Algebraic independence of homomorphisms", " 12. The normal basis theorem", "Chapter IX, Extensions of Rings", " 1. Integral ring extensions", " 2. Integral Galois extensions", " 3. Extension of homomorphisms", "Chapter X, Transcendental Extensions", " 1. Transcendence bases", " 2. Hilbert's Nullstellensatz", " 3. Algebraic sets", " 4. Noether normalization theorem", " 5. Linearly disjoint extensions", " 6. Separable extensions", " 7. Derivations", "Chapter XI, Real Fields", " 1. Ordered fields", " 2. Real fields", " 3. Real zeros and homomorphisms", "Chapter XII, Absolute Values", " 1. Definitions, dependence, and independence", " 2. Completions", " 3. Finite extensions", " 4. Valuations", " 5. Completions and valuations", " 6. Discrete valuations", " 7. Zeros of polynomials overcomplete fields", "Part Three, Linear Algebra and Representations", "Chapter XIII, Matrices and Linear Maps", " 1. Matrices", " 2. The rank of a matrix", " 3. Matrices and linear maps", " 4. Determinants", " 5. Duality", " 6. Matrices and bilinear forms", " 7. Sesquilinear duality", "Chapter XIV, Structure of Bilinear Forms", " 1. Preliminaries, orthogonal sums", " 2. Quadratic maps", " 3. Symmetric forms, orthogonal bases", " 4. Hyperbolic spaces", " 5. Witt's theorem", " 6. The Witt group", " 7. Symmetric forms over ordered fields", " 8. The Clifford algebra", " 9. Alternating forms", " 10. The Pfaffian", " 11. Hermitian forms", " 12. The spectral theorem (hermitian case)", " 13. The spectral theorem (symmetric case)", "Chapter XV, Representation of One Endomorphism", " 1. Representations", " 2. Modules over principal rings", " 3. Decomposition over one endomorphism", " 4. The characteristic polynomial", "Chapter XVI, Multilinear Products", " 1. Tensor product", " 2. Basic properties", " 3. Extension of the base", " 4. Tensor product of algebras", " 5. The tensor algebra of a module", " 6. The Alternating Product", " 7. Symmetric products", " 8. The Euler-Grothendieck Ring", " 9. Some functorial isomorphisms", "Chapter XVII, Semisimplicity", " 1. Matrices and linear maps over non-commutative rings", " 2. Conditions defining semisimplicity", " 3. The density theorem", " 4. Semisimple rings", " 5. Simple rings", " 6. Balanced modules", "Chapter XVIII, Representations of Finite Groups", " 1. Semisimplicity of the group algebra", " 2. Characters", " 3. One-dimensional representations", " 4. The space of class functions", " 5. Orthogonality relations", " 6. Induced characters", " 7. Induced representations", " 8. Positive decomposition of the regular character", " 9. Supersolvable groups", " 10. Brauer's theorem", " 11. Field of definition of a representation", "Appendix. The Transcendence of e and π", "Index".
Meget kompakt gennemgang af algebra på universitetsniveau. Stort set ingen eksempler i denne udgave. ( )
  bnielsen | Aug 26, 2013 |
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Amazon.com Product Description (ISBN 038795385X, Hardcover)

This book is intended as a basic text for a one year course in algebra at the graduate level or as a useful reference for mathematicians and professionals who use higher-level algebra. This book successfully addresses all of the basic concepts of algebra. For the new edition, the author has added exercises and made numerous corrections to the text. From MathSciNet's review of the first edition: "The author has an impressive knack for presenting the important and interesting ideas of algebra in just the "right" way, and he never gets bogged down in the dry formalism which pervades some parts of algebra."

(retrieved from Amazon Mon, 30 Sep 2013 13:48:52 -0400)

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